Large language models (LLMs) now exhibit near human-level performance in various tasks, but their performance drops drastically after a handful of high-resource languages due to the imbalance in pre-training data. Inspired by the human process of second language acquisition, particularly code-switching$\unicode{x2014}$the practice of language alternation in a conversation$\unicode{x2014}$we propose code-switching curriculum learning (CSCL) to enhance cross-lingual transfer for LLMs. CSCL mimics the stages of human language learning by progressively training models with a curriculum consisting of 1) token-level code-switching, 2) sentence-level code-switching, and 3) monolingual corpora. Using Qwen 2 as our underlying model, we demonstrate the efficacy of the CSCL in improving language transfer to Korean, achieving significant performance gains compared to monolingual continual pre-training methods. Ablation studies reveal that both token- and sentence-level code-switching significantly enhance cross-lingual transfer and that curriculum learning amplifies these effects. We also extend our findings into various languages, including Japanese (high-resource) and Indonesian (low-resource), and using two additional models (Gemma 2 and Phi 3.5). We further show that CSCL mitigates spurious correlations between language resources and safety alignment, presenting a robust, efficient framework for more equitable language transfer in LLMs. We observe that CSCL is effective for low-resource settings where high-quality, monolingual corpora for language transfer are hardly available.